Purely from a field theory perspective, can anyone explain through plain intuition why solvability of a polynomial implies the existence of a chain of normal extensions of a field than just extensions?
2026-03-28 06:07:01.1774678021
Why solvability implies sequence of normal extensions?
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First, any polynomial corresponds to a chain of extensions, but if those extensions are not "radical" extensions, then you can't express the roots of the polynomial in terms of radicals. So, solvability implies the existence of a chain of radical extensions. But every chain of radical extensions can be expanded to a chain of normal extensions by doing as Wojowu writes in a comment: first, toss in all the roots of unity you need, then toss in all the radicals you need.